Question: eV level of orbiting electrons

I've been reading of the Bohr model lately and I was wondering.....do we only know the energy levels of electrons orbiting hydrogen or do we know these levels for other atoms? If not, is there a way to calculate this using the Shrodinger or is this just part of the whole mystery?

In principle one can calculate the energy levels of all atoms. The only problem is that with growing number of electrons the numerical complexity increases. I don't know for which atoms this has been done explicitly.

The calculational tool is the Schroedinger equation for multi-electron systems. Of course one has to take into account corrections like spin-orbit. I guess that for heavy atoms / inner shells relativistic corrections will be included as well.

We we definitely know the energy levels from experiments. That's how astrophysicists can use spectra to determine the chemical composition of stars. But I'm guessing you're talking about explicitly calculating the energy levels. As Tom mentioned, the problem isn't theoretical but computational. Even for very simple systems with multiple particles, you run into integrals that are basically impossible to do. I guess the only other atomic system that can be explicitly calculated is the hydrogenic atom, i.e. an ion with just one electron. However, for simple elements like helium, you can use perturbation theory to get a pretty good approximation.

There are only numerical solutions to the Schrödinger equation, so the question is:
To what accuracy?

Certainly every element has been calculated to some accuracy by now; it only takes a minute or two to do a small-basis-set Hartree-Fock calculation on even the heaviest elements.
(And heavy elements are heavy, if you put into the perspective that a single plutonium atom has the same number of electrons as http://en.wikipedia.org/wiki/Anthracene" [Broken], a fairly big organic molecule!

For helium, in order, the approximations/corrections would be:
Basis set approximation, the approximations made by your method of solving the electronic Schrödinger equation, the Born-Oppenheimer approximation (treating the nucleus as stationary, allowing you to separate it into an electronic S.E.), relativistic mass corrections to the electron, spin-orbit effects, nuclear-spin-orbit effects, QED effects.

Now, we know these levels from spectroscopy primarily, because spectroscopy is relatively simple and accurate compared to a high-level QM calculations. E.g. a spectroscopic accuracy can be around 0.1 cm-1 which is about 1/1000 of a kJ/mol. But spectroscopy measures the difference between two levels, whereas QM requires calculating the total electronic energy. Which is on the order of thousands of kJ/mol. So it's not that these approximations are bad approximations (in terms of total electronic energy), it's just that to get a comparable level of accuracy, your QM calculation has to be 99.9999% accurate!

For which I have to disagree with arunma, first-order PT isn't much of an improvement even for helium. It's a nice and common example of PT for textbooks, but there are even http://arxiv4.library.cornell.edu/abs/0704.3549v1" [Broken] models which outperform it for simple two-electron systems like Helium.

The first really accurate quantum calculation on Helium (which you can do by hand, albeit tediously) was done by Hylleraas in 1929, who got the ground-state energy to 4 digits of accuracy. (99.99%, and by increasing the basis set, it's been calculated by the same method to truly http://arxiv.org/abs/math-ph/0605018" [Broken] was published where the system He2 was studied to a new level of accuracy! This is Science - there's always room to expand our knowledge!

So in short, we do believe we have all the necessary basic theory to calculate the spectroscopic and chemical properties of all elements and molecules to within experimental error. But the calculations in question are so complicated that we've yet to do so for every element, much less most chemical compounds!